Four masses of $2 \ kg$ each are attached to an axle by spokes of length $1/4 \ m$ as shown in the figure. $A$ force $F = 24 \ N$ is applied at the end of a spoke of length $1/2 \ m$ at an angle of $30^{\circ}$ to produce an angular acceleration $\alpha$. The value of $\alpha$ is ........ $rad/s^2$.

$A$ crew of scientists has built a new space station. The space station is shaped like a wheel of radius $R,$ with essentially all its mass $M$ at the rim. When the crew arrives,the station will be set rotating at a rate that causes an object at the rim to have radial acceleration $g,$ thereby simulating Earth's surface gravity. This is accomplished by two small rockets,each with thrust $T$ newtons,mounted on the station's rim. How long a time $t$ does one need to fire the rockets to achieve the desired condition?

$A$ flywheel of moment of inertia $0.4 \, kg \, m^2$ and radius $0.2 \, m$ is free to rotate about a central axis. If a string is wrapped around it and it is pulled with a force of $10 \, N$,then the angular velocity after $4 \, s$ will be ......... $rad \, s^{-1}$.

If there is a change of angular momentum from $1\,J\cdot s$ to $5\,J\cdot s$ in $5\,s$,then the torque is:

$A$ constant torque of $400 \ Nm$ acts on a wheel having a moment of inertia of $100 \ kg \ m^2$ about its central axis. The angular velocity acquired in $4 \ s$ is ....... $rad \ s^{-1}$.

The instantaneous angular position of a rotating wheel is given by the formula $\theta (t) = 2t^3 - 6t^2$. At what time will the torque on this wheel be zero? $t = \dots \text{s}$